solve for x 8 x22 x4 4x4 3 x 4 24 x2 16 x3 x2 4x 12 x

solve for x:
8- (x-2)/2= x/4

4/(x+4) + 3/ (x - 4) = 24/ (x^2 - 16)

(x+3)/ (x^2 - 4x - 12) + (x - 11)/ (x^2 - 2x - 24) = (x + 1) / (x^2 + 6x +8)

Solution

1.   8- (x-2)/2= x/4 or, [8*2 –(x-2)]/2 = x/4 or, 4(16-x+2) = 2x or, 72- 4x = 2x or, 6x = 72 so that x = 72/6 = 12.

2.   4/(x+4)+3/(x-4) = 24/(x²-16) or, [4(x-4)+3(x+4)]/(x+4)(x-4) = 24/(x² -16) or, (4x-16+3x+12)/(x² -16) = 24/(x²-16) or, (7x-4) )/(x² -16) = 24/(x² -16). Now, assuming that x²-160, i.e. x±4, in which case x²-16 = 0 and division by 0 is not defined), on multiplying both the sides by x² -16, we have 7x-4 = 24 or, 7x = 24+4 or, 7x = 28 so that x = 28/7 or, x = 4. Hence in this case ,there is no solution as division by x²-16 is not defined.

3.   (x+3)/ (x² - 4x - 12) + (x - 11)/ (x² - 2x - 24) = (x + 1) / (x² + 6x +8)

We have x² - 4x – 12 = x² - 6x +2x – 12 = x(x-6)+2(x-6) = (x-6)(x+2)

Also, x² - 2x – 24 = x² - 6x+4x – 24 = x(x-6)+4(x-6) = (x-6)(x+4) and

  x² + 6x +8 =x² + 2x+4x +8 = x(x+2) +4(x+2)= (x+2)(x+4).

Hence the given expression changes to (x+3)/(x-6)(x+2)+ (x-11)/(x-6)(x+4) = (x+1)/(x+2)(x+4) or,

(x+3)/(x-6)(x+2)+(x-11)/(x-6)(x+4) - (x+1)/(x+2)(x+4)= 0   or, [(x+3)(x+4)+(x-11)(x+2)-(x+1)(x-6)] / (x-6)(x+2)(x+4) = 0 or,

( x² +7x+12 +x²-9x -22- x² +5x+ 6) /(x-6)(x+2)(x+4) = 0 or,

(2x²+3x- 4) /(x-6)(x+2)(x+4) = 0. Now, assuming that x 6,or – 2 or -4, ( in which case (x-6)(x+2)(x+4) = 0 and division by 0 is not defined), on multiplying both the sides by (x-6)(x+2)(x+4), we get 2x² +3x-4 = 0. Further, on using the quadratic formula, we get x = [-(3)±{3²- 4*2*(-4)}]/2*2 or x = [3±(9 +32)]/4 or, x = [3±41]/4.